Expected Number of Vertices of a Random Convex Polytope. I. Integral Formula and Asymptotic Bounds

摘要

Given m points on the unit sphere in n-space, the hyperplanes tangent to the sphere at the given points bound a convex polytope with m facets. If the points are chosen independently at random from the uniform distribution on the sphere, the number V sub mn of the vertices of the polytope is a random variable. We obtain an integral expression for EV sub mn and asymptotic bounds of the form alpha to the n power n to the (n-6)/2 power (M-N) < or = EV sub mn < or = beta to the n power n to the (n-5)/2 power. (Author)